Abstract
We study the partial capacitated vertex cover problem (pcvc) in which the input consists of a graph G and a covering requirement L. Each edge e in G is associated with a demand ℓ(e), and each vertex v is associated with a capacity c(v) and a weight w(v). A feasible solution is an assignment of edges to vertices such that the total demand of assigned edges is at least L. The weight of a solution is ∑ v α(v) w(v), where α(v) is the number of copies of v required to cover the demand of the edges that are assigned to v. The goal is to find a solution of minimum weight. We consider three variants of pcvc. In pcvc with separable demands the only requirement is that total demand of edges assigned to v is at most α(v) c(v). In pcvc with inseparable demands there is an additional requirement that if an edge is assigned to v then it must be assigned to one of its copies. The third variant is the unit demands version. We present 3-approximation algorithms for both pcvc with separable demands and pcvc with inseparable demands and a 2-approximation algorithm for pcvc with unit demands. We show that similar results can be obtained for pcvc in hypergraphs and for the prize collecting version of capacitated vertex cover. Our algorithms are based on a unified approach for designing and analyzing approximation algorithms for capacitated covering problems. This approach yields simple algorithms whose analyses rely on the local ratio technique and sophisticated charging schemes.
Research supported in part by REMON — Israel 4G Mobile Consortium.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Karp, R.M.: Reducibility among combinatorial problems. Complexity of Computer Computations, 85–103 (1972)
Dinur, I., Safra, S.: The importance of being biased. In: 34th ACM Symp. on the Theory of Computing, pp. 33–42 (2002)
Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problem. PWS Publishing Company (1997)
Bar-Yehuda, R., Bendel, K., Freund, A., Rawitz, D.: Local ratio: a unified framework for approximation algorithms. ACM Comp. Surveys 36(4), 422–463 (2004)
Guha, S., Hassin, R., Khuller, S., Or, E.: Capacitated vertex covering. Journal of Algorithms 48(1), 257–270 (2003)
Lustbader, J.W., Puett, D., Ruddon, R.W. (eds.): Symposium on Glycoprotein Hormones: Structure, Function, and Clinical Implications (1993)
Wolsey, L.A.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2, 385–393 (1982)
Gandhi, R., Khuller, S., Parthasarathy, S., Srinivasan, A.: Dependent rounding and its applications to approximation algorithms. Journal of the ACM 53(3), 324–360 (2006)
Chuzhoy, J., Naor, J.: Covering problems with hard capacities. In: 43nd IEEE Symp. on Foundations of Comp.Sci., pp. 481–489 (2002)
Gandhi, R., Halperin, E., Khuller, S., Kortsarz, G., Srinivasan, A.: An improved approximation algorithm for vertex cover with hard capacities. Journal of Computer and System Sciences 72(1), 16–33 (2006)
Kearns, M.: The Computational Complexity of Machine Learning. MIT Press, Cambridge (1990)
Slavík, P.: Improved performance of the greedy algorithm for partial cover. Information Processing Letters 64(5), 251–254 (1997)
Bshouty, N.H., Burroughs, L.: Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 298–308. Springer, Heidelberg (1998)
Bar-Yehuda.: Using homogeneous weights for approximating the partial cover problem. Journal of Algorithms 39, 137–144 (2001)
Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. Journal of Algorithms 53(1), 55–84 (2004)
Srinivasan, A.: Distributions on level-sets with applications to approximation algorithms. In: 42nd IEEE Symp. on Foundations of Comp. Sci., pp. 588–597 (2001)
Halperin, E., Srinivasan, A.: Improved approximation algorithms for the partial vertex cover problem. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 161–174. Springer, Heidelberg (2002)
Mestre, J.: A primal-dual approximation algorithm for partial vertex cover: Making educated guesses. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 182–191. Springer, Heidelberg (2005)
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–46 (1985)
Bar-Yehuda, R.: One for the price of two: A unified approach for approximating covering problems. Algorithmica 27(2), 131–144 (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bar-Yehuda, R., Flysher, G., Mestre, J., Rawitz, D. (2007). Approximation of Partial Capacitated Vertex Cover . In: Arge, L., Hoffmann, M., Welzl, E. (eds) Algorithms – ESA 2007. ESA 2007. Lecture Notes in Computer Science, vol 4698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75520-3_31
Download citation
DOI: https://doi.org/10.1007/978-3-540-75520-3_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75519-7
Online ISBN: 978-3-540-75520-3
eBook Packages: Computer ScienceComputer Science (R0)